\subsection{Renomalization of two-channel model (more)-study of Jackson's paper}
Eq. (9) in the paper \cite{JacksonNarrow} gives U.  Presumably,  $G^{vac}$ is the same as $G_c$ or $G_o$, therefore, $U$ is already renormalized for high-k. And $\Delta{G_{o,c}}$ in eq. (12) is cut off at high-k. 

